If an airline is using an airport cheaply and said airport continuously requires government money to not go bankrupt, that is a fairly clear subsidy. The "rent retail space and chance at attracting other carriers" doesn't work that reliably in practice, since those other carriers can play the same game of "we only come if we don't have to pay the actual costs", and can quickly shift to other airports if one stops playing ball. Sometimes it seems attracting airplane maintenance facilities is the most reliable source of income for such airports.
Especially since in many areas, long-distance air travel is pretty much a zero sum game: more small airports doesn't mean more people flying, it's just shifting people around between airports.
In truly remote areas I believe that's different and worth supporting (paying for infrastructure so people can travel more easily, when it'd otherwise be a day or two in the car to get to an airport), but e.g. in Germany there's a bunch of places that have good train connections to bigger airports, but regional airports with extremely limited air connections are still funded. E.g. one example I'm thinking of is less than 2 hours by train from 2 big airports, but apparently needs an airport that serves a few holiday flights each week. Every few years there's big announcements of new airlines coming in, which either get cancelled again after a few years, the airline goes bankrupt or ... While the running subsidies aren't that high, the initial investments were and IMHO would have been better used for other infrastructure.
"If an airline is using an airport cheaply and said airport continuously requires government money to not go bankrupt, that is a fairly clear subsidy."
Yes. But based on EU regulation this won't be allowed anymore in the near future. (2020? 2022?)
Especially since in many areas, long-distance air travel is pretty much a zero sum game: more small airports doesn't mean more people flying, it's just shifting people around between airports.
In truly remote areas I believe that's different and worth supporting (paying for infrastructure so people can travel more easily, when it'd otherwise be a day or two in the car to get to an airport), but e.g. in Germany there's a bunch of places that have good train connections to bigger airports, but regional airports with extremely limited air connections are still funded. E.g. one example I'm thinking of is less than 2 hours by train from 2 big airports, but apparently needs an airport that serves a few holiday flights each week. Every few years there's big announcements of new airlines coming in, which either get cancelled again after a few years, the airline goes bankrupt or ... While the running subsidies aren't that high, the initial investments were and IMHO would have been better used for other infrastructure.